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Pantelis E. Eleftheriou

Publications and source records attributed to Pantelis E. Eleftheriou.

18 recordsLinked to original sources
arXiv2026

On the global linear Zarankiewicz problem

The `global' Zarankiewicz problem for hypergraphs asks for an upper bound on the number of edges of a finite $r$-hypergraph $V$ in terms of the number $|V|$ of its vertices, assuming the edge relation is induced by a fixed $K_{k, \dots, k}$-free $r$-hypergraph $E$, for some $k\in\mathbb N$. In [4], such bounds of size $O(|V|^{r-1})$ were achieved for a semilinear $E$, namely, definable in a linear o-minimal structure. We establish the same bounds in five new settings: when $E$ is definable in (a) a semibounded o-minimal structure and the vertex set of $V$ is `sufficiently distant', (b) a model of Presburger arithmetic, (c) the expansion $\langle\mathbb R,<,+, \mathbb Z\rangle$ of the real ordered group by the set of integers, (d) a stable 1-based structure without the finite cover property, and (e) a locally modular regular type in a stable theory, such as the generic type of the solution set of the Heat differential equation. Our methods include techniques for reducing Zarankiewicz's problem to the setting of arbitrary subgroups of powers of groups, used in geometric cases (a)--(c). They also include an abstract version of Zarankiewicz's problem for general saturated `linear structures' that yields the desired bounds in the model-theoretic settings (d)--(e), as well as a parametric version in (b). Furthermore, the bounds in (a) characterise those o-minimal structures that do not recover a global field, and in (c) they yield new versions of Zarankiewicz's problem for certain ordered abelian groups.

math.LO↗
arXiv2025

Orthogonality and domination in o-minimal expansions of ordered groups

We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not.

math.LO↗
arXiv2025

Exact VC-Dimensions of Certain Geometric Set Systems

The VC-dimension of a family of sets is a measure of its combinatorial complexity used in machine learning theory, computational geometry, and even model theory. Computing the VC-dimension of the $k$-fold union of geometric set systems has been an open and difficult combinatorial problem, dating back to Blumer, Ehrenfeucht, Haussler, and Warmuth in 1989, who ask about the VC-dimension of $k$-fold unions of half-spaces in $\mathbb{R}^d$. Let $\mathcal{F}_1$ denote the family of all lines in $\mathbb{R}^2$. It is well-known that $\mathsf{VC}\text{-}\mathsf{dim}(\mathcal{F}_1) = 2$. In this paper, we study the $2$-fold and $3$-fold unions of $\mathcal{F}_1$, denoted $\mathcal{F}_2$ and $\mathcal{F}_3$, respectively. We show that $\mathsf{VC}\text{-}\mathsf{dim}(\mathcal{F}_2) = 5$ and $\mathsf{VC}\text{-}\mathsf{dim}(\mathcal{F}_3) = 9$. Moreover, we give complete characterisations of the subsets of $\mathbb{R}^2$ of maximal size that can be shattered by $\mathcal{F}_2$ and $\mathcal{F}_3$, showing they are exactly two and five, respectively, up to isomorphism in the language of the point-line incidence relation.

math.CO↗
arXiv2022

Orthogonal decomposition of definable groups

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. Finally, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

math.LO↗
arXiv2021

On semibounded expansions of ordered groups

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle R, <, +, \dots\rangle$ is a semibounded o-minimal structure and $P\subseteq R$ a set satisfying certain tameness conditions, then $\langle \cal R, P\rangle$ remains semibounded. Examples include the cases when $\mathcal{R}=\langle \mathbb R,<,+, (x\mapsto λx)_{λ\in \mathbb R}, \cdot_{ [0, 1]^2} \rangle$, and $P= 2^\mathbb Z$ or $P$ is an iteration sequence. As an application, we obtain that smooth functions definable in such $\langle \mathcal R, P\rangle$ are definable in $\mathcal R$.

math.LO↗
arXiv2020

Coincidence of dimensions in closed ordered differential fields

Let $\mathcal K=\langle\mathcal R, δ\rangle$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal M=\langle \mathcal R, C\rangle$, the $δ$-dimension and the large dimension coincide. As an application, we characterize the definable sets in $\mathcal K$ that are internal to $C$ as those sets that are definable in $\mathcal M$ and have $δ$-dimension $0$. We further show that, for sets definable in $\mathcal K$, having $δ$-dimension $0$ does not generally imply co-analyzability in $C$ (in contrast to the case of transseries). We also point out that the coincidence of dimensions also holds in the context of differentially closed fields and in the context of transseries.

math.LO↗
arXiv2020

Groups definable in weakly o-minimal non-valuational structures

Let $\mathcal M$ be a weakly o-minimal non-valuational structure, and $\mathcal N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal M$ is a subgroup of a group $K$ definable in $\mathcal N$, which is canonical in the sense that it is the smallest such group. As an application, we obtain that $G^{00}= G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if $G$ has finitely satisfiable generics, then $\dim_{Lie}(G/G^{00})= \dim(G)$.

math.LO↗
arXiv2019

Structure theorems in tame expansions of o-minimal structures by a dense set

We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.

math.LO↗
arXiv2019

Semi-linear stars are contractible

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The proof goes through the stronger statement that the star of a cell in a special linear decomposition of $X$ is definably simply-connected. In fact, if the star is bounded, then it is definably contractible.

math.LO↗
arXiv2019

Locally definable subgroups of semialgebraic groups

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

math.LO↗
arXiv2018

Expansions of real closed fields which introduce no new smooth functions

We prove the following theorem: let $\widetilde{\mathcal R}$ be an expansion of the real field $\overline{\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a "semialgebraic chunk". Then every definable smooth function $f:X\subseteq \mathbb R^n\to \mathbb R$ with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions $\widetilde{\mathcal R} = \langle \overline{\mathbb R}, P\rangle$ of the real field, such as when $P=2^\mathbb Z$, or $P\subseteq \mathbb R$ is an iteration sequence. A generalization of the theorem to d-minimal expansions $\widetilde{\mathcal R}$ of $\mathbb R_{an}$ fails. On the other hand, we prove our theorem for expansions$\widetilde{\mathcal R}$ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as $\langle \overline{\mathbb R}, \mathbb R_{alg}, 2^\mathbb Z\rangle$.

math.LO↗
arXiv2018

Small sets in dense pairs

Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that the induced structure on $P$ by $\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set $X$ definable in $\widetilde{\mathcal M}$ can be definably embedded into some $P^l$, uniformly in parameters, settling a question from [10]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of $\mathcal M$ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [17]. The above results are in contrast to recent literature, as it is known in general that $\widetilde{\mathcal M}$ does not eliminate imaginaries, and neither it nor the induced structure on $P$ admits definable Skolem functions.

math.LO↗
arXiv2018

Small sets in Mann pairs

Let $\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$ be an expansion of a real closed field $\mathcal M$ by a dense subgroup $G$ of $\langle M^{>0}, \cdot\rangle$ with the Mann property. We prove that the induced structure on $G$ by $\mathcal M$ eliminates imaginaries. As a consequence, every small set $X$ definable in $\mathcal M$ can be definably embedded into some $G^l$, uniformly in parameters. These results are proved in a more general setting, where $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ is an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, satisfying three tameness conditions.

math.LO↗
arXiv2018

Characterizing o-minimal groups in tame expansions of o-minimal structures

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of $\mathcal M$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely their dimension equals the dimension of their topological closure, (3) if $\mathcal N$ expands $\mathcal M$ by a dense independent set, then every definable group is o-minimal.

math.LO↗
arXiv2018

Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field.

math.LO↗
arXiv2017

Product cones in dense pairs

Let $\mathcal M=\langle M, <, +, \dots\rangle$ be an o-minimal expansion of an ordered group, and $P\subseteq M$ a dense set such that certain tameness conditions hold. We introduce the notion of a `product cone' in $\widetilde{\mathcal M}=\langle \cal M, P\rangle$, and prove: if $\mathcal M$ expands a real closed field, then $\widetilde{\mathcal M}$ admits a product cone decomposition. If $\mathcal M$ is linear, then it does not. In particular, we settle a question from [10].

math.LO↗
arXiv2016

On definable Skolem functions in weakly o-minimal non-valuational structures

We prove that all known examples of weakly o-minimal non-valuational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to (definable families of) definable cuts. Along the way we give some new examples of weakly o-minimal non-valuational structures.

math.LO↗